year 7 Alg Pilot 6-10

A1: notation

A2: substitution

A3: concepts and
vocabulary

A4: Simplification
and manipulation

A6: modelling with
algebra

Lessons 6-10 overview

Lesson 6: Collecting like terms

term

like terms

constant

multiple

Simplify:

2𝑚 + 𝑚 + 3𝑚
3𝑐 + 1 + 4𝑐 − 2

3𝑝𝑞 + 7 + 2𝑝𝑞 − 3

Linear terms only:
Not e.g. 2𝑥2+ 3𝑥 + 4𝑥2− 5𝑥

Lesson 7: Expanding brackets

expanding

Expand:

3 4𝑥 + 2
−4 𝑎 − 5
𝑏(3𝑏 − 5)
3𝑎(2𝑎 + 4)

Lesson 8: Expanding and

simplifying

Expand and simplify:

3 𝑎 + 2 + 4(𝑎 − 5)

3 3𝑏 − 1 + 4(2𝑏 − 3)

3 𝑎 + 2 − 4(𝑎 + 5)

3 3𝑏 − 1 − 4(2𝑏 − 3)

Linear terms only:
NOT e.g. 𝑏 3𝑏 − 1 + 𝑏(2𝑏 + 6)

Lesson 9: Algebraic factors

List all the factors of 5𝑎𝑏
List all the factors of 21𝑝𝑞
List all the common factors of 3𝑎𝑏 and 9𝑏𝑐
HCF of 4𝑎 and 12
HCF of 4𝑎2 and 12𝑎

Lesson 10: Factorising

factorise

fully factorise

Factorise 3𝑥 + 6
Factorise 6𝑎 + 7𝑎2

Fully factorise 12b + 18
Fully factorise 5𝑎 − 15𝑎2

Students will:

Example

understand that like terms are those that are constants, multiples
of the same letter, and multiples of the same product

1 and 5 are like terms because they are constants
2𝑥 and 3𝑥 are like terms because they are multiples of 𝑥
4𝑎𝑏 and 5𝑎𝑏 are like terms because they are multiples of 𝑎𝑏

understand that each term has a positive or negative value

The terms in 7𝑎 + 5 − 5𝑎 − 2 are 7𝑎, +5, −5𝑎, and −2

be able to simplify expressions with one set of like terms
Simplify 2𝑚 + 𝑚 + 3𝑚
Simplify 5𝑏 − 3𝑏

be able to simplify expressions with multiple sets of like terms
Simplify 3c + 1 + 4c − 2
Simplify 3𝑝𝑞 + 7 + 2𝑝𝑞 − 3

Lesson 6: Collecting like terms

Unit

Example

Y7U2: Properties of
arithmetic

Distributivity – expanding brackets with
numbers

Y7U3: Factors and
multiples
Factors and factor pairs

Y7U4: Prime Factor
Decomposition
Highest common factors

Prior learning:

Learning objectives:

Retrieve

Antoni, Binh and James order some food.

How much money did they spend

altogether?

Find a different way of working it out.

describe

Retrieve - ANSWERS

Instead of adding up the items individually,

we can group the items with the same value:

describe

Two lots of fruit
salad = 2 × £4

Three bottles of
water = 3 × £2

Two wraps =

2 × £6

£8 + £6 + £12 = £26

£8

£6

£12

Explain

5 + 5 + 5 + 5 = 4 × 5

We can use grouping to simplify calculations:

This is the same

as 4 lots of 5

𝑎 + 𝑎 + 𝑎 = 3 × 𝑎 = 3𝑎

This is the same

as 3 lots of 𝑎

𝑎

𝑎+

𝑎+

𝑎

𝑎

𝑎

=

+

+

+

=

Model

2𝑚 + 𝑚 + 3𝑚

Write these expressions as single terms of 𝑚

7𝑚 − 3𝑚

𝑚

+

𝑚 + 𝑚

𝑚 𝑚 𝑚

𝑚 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚

−𝑚 −𝑚 −𝑚

A

© Copyright text
Quick Check

A

B

D

C

𝑒4

4𝑒

𝑒4

𝑒 × 4

Simplify using algebraic conventions:

𝑒 + 𝑒 + 𝑒 + 𝑒

© Copyright text
Quick Check

A

B

D

C

3𝑝

6𝑝

7𝑝

𝑝7

Write as a single term of 𝑝:

4𝑝 + 2𝑝 + 𝑝

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Quick Check

A

B

D

C

2𝑝

3𝑝

Write as a single term of 𝑝:

5𝑝 − 3𝑝 + 2𝑝

4𝑝

𝑝2

A

Explain

5 + 5 + 3 + 5 + 3 = 3 × 5 + 2 × 3

We can group the 5s and

group the 3s in this expression:

𝑎 + 𝑎 + 𝑏 + 𝑎 + 𝑏 = 3 × 𝑎 + 2 × 𝑏

= 3𝑎 + 2𝑏

𝑎 and 𝑏 represent
different numbers.

=

+

+

+

=+

𝑎 +

+

𝑎 +

𝑎 +

𝑏

𝑏

𝑎𝑎𝑎+ 𝑏

𝑏

=

We can group the 𝑎s and

group the 𝑏s in this expression:

Explain

A term is something being added or subtracted in an expression.

A term can be a number, a letter or a product of numbers and letters.

The sign in front of the term is part of the term.

4𝑎 + 3 − 6𝑎𝑏

This is a 3-term

expression.

The terms are:

4𝑎, +3 and −6𝑎𝑏

This is a …-term

expression.

The terms are:

…………………………

−2 + 14𝑟 − 4𝑡𝑦2+ 9

Explain

Multiples of the same letter are called like terms.

The sign in front of the term is part of the term.

3𝑎 + 4𝑏 + 𝑎 − 5𝑏

multiple of 𝑎
multiple of 𝑏

3𝑎 and 𝑎 are like terms
because they are both

multiples of 𝒂.

… and … are like terms
because they are both

multiples of … .

Model

What are the like terms in these expressions?

4𝑟 + 3𝑠 + 𝑡 + 2𝑟 + 3𝑡 + 𝑠

3𝑒 − 𝑓 + 𝑒 + 2𝑓

We can circle or box like

terms in the same colour or

shape to help us simplify.
Remember to include the

sign in front of the term.

B

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Quick Check

A

B

D

C

2𝑥 and 3𝑦

2𝑥 and −2

−2 and −2𝑦

+3𝑦 and −2𝑦

Which are the like terms in the expression:

2𝑥 + 3𝑦 − 2 − 2𝑦

?

Model

We can simplify expressions by collectingliketerms.

2𝑎 + 𝑏 + 𝑎 + 3𝑏

𝑎

𝑎

𝑏

𝑏

𝑏
𝑎

𝑏

𝑎

𝑎

𝑎

𝑏

𝑏

𝑏

𝑏

6𝑎 + 2𝑏 − 2𝑎 − 𝑏

𝑎

𝑎

𝑎

𝑎

𝑎

𝑎

𝑏

𝑏

−𝑎

−𝑎
−𝑏

𝑎

𝑎

𝑎

𝑎

𝑎

𝑎

−𝑎

−𝑎

𝑏

𝑏

−𝑏

𝑎

𝑎

𝑎

𝑎
𝑏

Model

Simplify these expressions by collecting like terms.

4𝑟 + 3𝑠 + 𝑡 + 2𝑟 + 3𝑡 + 𝑠

3𝑒 − 𝑓 + 𝑒 + 2𝑓

C

© Copyright text
Quick Check

A

B

D

C

5𝑐 + 5𝑑

10𝑐𝑑

9𝑐2𝑑2

2𝑐 + 2𝑑

Simplify by collecting like terms:

4𝑐 + 3𝑑 + 𝑐 + 2𝑑

© Copyright text
Quick Check

A

B

D

C

9𝑚𝑛

6𝑚 + 𝑛

6𝑚 − 2

6𝑚 − 𝑛

Simplify by collecting like terms:

5𝑚 + 𝑛 + 𝑚 − 2𝑛

© Copyright text
Ready to go?

C

Simplify by collecting like terms:

2𝑎 − 5𝑏 + 4𝑎 + 2𝑏

Explain

Constants are terms without any variables; they are just numbers.

Constants are like terms.

Multiples of the same combinations of letters are like terms.

4𝑐𝑑 − 3 − 2𝑐𝑑 + 6

multiple of 𝑐𝑑
constant

… and … are like

terms because they

are both multiples of …

… and … are like

terms because they

are both …

Model

Simplify these expressions by collecting like terms:

4𝑝𝑞 + 3 − 𝑝𝑞 − 8

7𝑚 + 2 − 5𝑚 + 12

D

© Copyright text
Ready to go?

Simplify

8𝑥 + 3 − 5𝑥 − 1

D

Talk Task

The Maths Mastery students are trying to simplify some expressions.

= 4x – x + 2
= 4 + 2
= 6

=3a + 2b + 5ab
= 10ab

Explain their mistakes.

spot the
mistake

= 4+5y+ 2y
= 9y + 2y
= 11y

= 5v + 2w - 4v + 3w

= 9v + 5w

Talk Task

The Maths Mastery students are trying to simplify some expressions.

= 4x – x + 2
= 4 + 2
= 6

=3a + 2b + 5ab
= 10ab

Explain their mistakes.

spot the
mistake

= 4+5y+ 2y
= 9y + 2y
= 11y

= 5v + 2w - 4v + 3w

= 9v + 5w

The student
thinks that
subtracting 𝑥
removes it
from 4𝑥

The student
has added
3𝑎 and 2𝑏

The student
has added 4
and 5𝑦

The student
hasn’t
noticed the
negative
sign of −4v

Purposeful Practice

What if you use
other variables?
Extension prompt

3

What about if you

use decimals?

Extension prompt

2

What if you use

negatives?

Extension prompt

1

𝑎 + 𝑏

2𝑎 + 3𝑏

3𝑎 + 4𝑏

The expression in the white box is found by

adding the expressions in the two blocks below it.

How many other expressions can you find

that could go in the pink and green boxes?

2𝑎 + 3𝑏+ 𝑎 + 𝑏

simplifies to 3𝑎 + 4𝑏.

Purposeful Practice

−3𝑎 + 3𝑏

6𝑎 − 𝑏

3𝑎 + 4𝑏

There are an infinite number of examples you could use:

e.g.

2.5𝑎 + 3.5𝑏

0.5𝑎 + 0.5𝑏

3𝑎 + 4𝑏

𝑎 + 𝑏 + 𝑐

2𝑎 + 3𝑏 − 𝑐

3𝑎 + 4𝑏

Exit Ticket

1) Simplify these expressions fully:

a) 𝑚 + 𝑚 + 𝑚 + 𝑚

b) 5𝑒 − 2𝑒 + 4𝑒

c) 3𝑥 + 5𝑦 + 2𝑥 − 2𝑦

d) 4𝑥 + 3 − 𝑥 + 5

e) 10 + 3𝑐 + 5𝑑 − 7𝑐 + 𝑑 + 4

Exit Ticket ANSWERS

1) Simplify these expressions fully:

a) 𝑚 + 𝑚 + 𝑚 + 𝑚

b) 5𝑒 − 2𝑒 + 4𝑒

c) 3𝑥 + 5𝑦 + 2𝑥 − 2𝑦

d) 4𝑥 + 3 − 𝑥 + 5

e) 10 + 3𝑐 + 5𝑑 − 7𝑐 + 𝑑 + 4

4𝑚

7𝑒

5𝑥 + 3𝑦

3𝑥 + 8

14 − 4𝑐 + 6𝑑

Students will:

Example

understand that a pair of brackets is multiplied by its coefficient

3(𝑥 + 2) means 3 × (𝑥 + 2)

understand that expanding a pair of brackets means everything
inside the brackets is multiplied by the coefficient (distributive law)

3 𝑥 + 2 = 3 × 𝑥 + [3 × 2]

3 𝑥 + 2 = 𝑥 + 2 + 𝑥 + 2 + [𝑥 + 2]

be able to multiply a single numerical term over a bracket
Expand 3(2𝑥 + 2)
Expand −4(3𝑎 − 5)

be able to multiply a single algebraic term over a bracket
Expand 𝑥(𝑥 + 2)
Expand 𝑎(2𝑎 − 5)

Lesson 7: Expanding Brackets

Unit

Example

Y7U2: Properties of
arithmetic

Distributivity – expanding brackets with
numbers

Y7U6: Positive and
negative numbers
Factors and factor pairs

Prior learning:

Learning objectives:

Retrieve

A mug weighs 400𝑔.

The box it comes in weighs 120𝑔.

Cala writes:

5 x (400 + 120)

Phil writes:

5 x 400 + 5 x 120

400𝑔

120𝑔

Phil and Cala are working out the total weight of

five of the mugs and their boxes.

Which of their methods will give the correct answer?

Is there another way to do it?

compare

Retrieve

5 x (400g + 120g)

5 x 400g + 5 x 120g

400𝑔

120𝑔

Both these calculations give the same answer:

compare

520𝑔
520𝑔
520𝑔
520𝑔
520𝑔

400𝑔
400𝑔
400𝑔
400𝑔
400𝑔

120𝑔
120𝑔
120𝑔
120𝑔
120𝑔

5 ×

𝟐𝟔𝟎𝟎𝒈

+ 5 ×

𝟐𝟔𝟎𝟎𝒈

2000𝑔

+

600𝑔

5 ×

Explain

3(2𝑎 + 1) means 3 × (2𝑎 + 1).

𝑎

1

𝑎

𝑎

1

𝑎

𝑎

1

𝑎

This is 3 lots
of (2𝑎 + 1)

6𝑎 + 3

Explain

3 lots of (2𝑎 + 1) is the same as

3 lots of 2𝑎and 3 lots of 1.

𝑎

1

𝑎

𝑎

1

𝑎

𝑎

1

𝑎

𝑎

1

𝑎

𝑎

1

𝑎

𝑎

1

𝑎

This is 3 lots
of (2𝑎 + 1)

This is 3 lots of
2𝑎 and 3 lots

of 1

6𝑎 + 3

6𝑎 + 3

3 × (2𝑎 + 1)

3 × 2𝑎 + 3 × 1

Model

Expanding a pair of brackets means multiplying the terms

inside the brackets by the coefficient of the brackets.

Every term inside the brackets is

multiplied by the number outside the brackets.

2 × (3𝑥 + 4)

2 × 3𝑥

2(3𝑥 + 4)

=

= 6𝑥 + 8

𝑥

1

𝑥

𝑥

1

1

1

𝑥

1

𝑥

𝑥

1

1

1

𝑥

𝑥

𝑥

𝑥

𝑥

𝑥

1

1

1

1

1

1

1

1

=

+ 2 × 4

Model

Expand these brackets:

2 3𝑥 + 4

3(7𝑥 − 4)

𝑥(𝑥 − 1)

×

We can use a multiplication grid to help us expand brackets.

×

𝑥
1
𝑥𝑥
1

1

1

𝑥
1
𝑥𝑥
1

1

1

×

3𝑥

+4

2

© Copyright text
Quick Check

A

B

D

C

−2

15

−15

2

What is the missing number?

×

2𝑚

−5

3

6𝑚

© Copyright text
Quick Check

A

B

D

C

7𝑐 + 6

12𝑐 + 8

12𝑐 + 6

12𝑐 + 2

Expand

4(3𝑐 + 2)

Talk Task

These students have expanded 8 2𝑝 − 3 , but none of their

answers are correct.

What mistakes have they made?

16p + 5

10p – 5

16p + 24

16p – 3

-8p

spot the
mistake

Talk Task

These students have expanded 8 2𝑝 − 3 , but none of their

answers are correct.

What mistakes have they made?

Subtracted 3 instead
of multiplying

Added the 8
and the 2𝑝

Forgot to multiply
second term

Didn’t notice it was
a negative 3

Confused like
terms. Either she
found the
correct answer
and then wrote
16𝑝 − 24 = −8𝑝
or she thought
2𝑝 − 3 = −1.

spot the
mistake

16p + 5

10p – 5

16p + 24

16p – 3

-8p

© Copyright text
Ready to go?

Expand:

5 2𝑐 + 3

Purposeful Practice

Zaki is trying to expand the bracket but has smudged his work.

−4(8 + 2𝑥 + 𝑦)

I know each smudge

was a + or a −.

How many possible correct answers could there be?

+

–

Purposeful Practice

For each 4 cards, there are 24 ways the numbers can be arranged e.g.:

−4(8 − 2𝑥 − 𝑦)

−4(8 − 2𝑥 + 𝑦)

−4(8 + 2𝑥 − 𝑦)

−4(8 + 2𝑥 + 𝑦)

+4(8 + 2𝑥 + 𝑦)

+4(8 + 2𝑥 − 𝑦)

+4(8 − 2𝑥 + 𝑦)

+4(8 − 2𝑥 − 𝑦)

When a number is

positive, we don’t always
need to write the + sign

Exit Ticket

×

𝑥

−7

5

5 𝑥 − 7 =

×

𝑥

+3

𝑥

𝑥 𝑥 + 3 =

2) Phil writes 3(4𝑥 − 8) = 12𝑥 − 8. Explain his mistake.

1) Use the grids to help you expand the brackets:

a) 5(𝑥 − 7)

b) 𝑥(𝑥 + 3)

Exit Ticket ANSWERS

×

𝑥

−7

5

5 𝑥 − 7 =

×

𝑥

+3

𝑥

𝑥 𝑥 + 3 =

2) Phil writes 3(4𝑥 − 8) = 12𝑥 − 8. Explain his mistake.

1) Use the grids to help you expand the brackets:

a) 5(𝑥 − 7)

b) 𝑥(𝑥 + 3)

5𝑥

−35

5𝑥 − 35

𝑥2+ 3𝑥

𝑥2

+3𝑥

Phil has multiplied 4𝑥 by 3 but he hasn’t multiplied −8 by 3.

Students will:

Example

understand that an expression can have more than one pair of
brackets
3 𝑥 + 4 + 5(𝑥 − 5)

understand that each pair of brackets has its own coefficient

3 𝑥 + 4 + 5 𝑥 − 5 = 3(𝑥 + 4) and 5 𝑥 − 5

understand that the coefficient of the pair of brackets can be
negative or positive
2 𝑥 + 1 − 3 𝑥 + 7 = +2(𝑥 + 1) and −3 𝑥 + 7

be able to expand and simplify expressions with positive
coefficients

3 𝑎 + 2 + 4 𝑎 − 5
3 3𝑏 + 1 + 4 𝑏 − 3

be able to expand and simplify expressions with negative
coefficients

3 𝑎 + 2 − 4 𝑎 + 5
3 3𝑏 + 1 − 4 𝑏 − 3

Lesson 8: Expanding and simplifying

Unit

Example

Y7U2: Properties of
arithmetic

Distributivity – expanding brackets with
numbers

Y7U3: Factors and
multiples
Factors and factor pairs

Y7U4: Prime Factor
Decomposition
Highest common factors

Prior learning:

Learning objectives:

Retrieve

A supermarket sells bags of fruit.
There are 2𝑎 − 3 apples in a bag.
There are 𝑎 + 1 oranges in a bag.

How many pieces of fruit do I have if I buy
5 bags of apples and 3 bags of oranges?

Can you write the expression in a different way?
compare

Retrieve - ANSWERS

Apples:

2𝑎 − 3 + 2𝑎 − 3 + 2𝑎 − 3 + 2𝑎 − 3 + 2𝑎 − 3 = 10𝑎 − 15

Oranges:

𝑎 + 1 + 𝑎 + 1 + 𝑎 + 1 = 3𝑎 + 3

Altogether:

10𝑎 − 15 + 3𝑎 + 3 = 𝟏𝟑𝒂 − 𝟏𝟐

A different way of writing this expression is:

5 2𝑎 − 3 + 3 𝑎 + 1

Explain

×

2𝑎

−3

5

10𝑎

−15

To simplify 5 2𝑎 − 3 + 3(𝑎 + 1) we need to

expand the brackets and then collect like terms.

5 2𝑎 − 3 + 3(𝑎 + 1)

×

𝑎

+1

+3

+3𝑎

+3

10𝑎 − 15

+3𝑎 + 3

= 10𝑎 − 15 + 3𝑎 + 3

= 10𝑎 + 3𝑎 − 15 + 3

= 13𝑎 − 12

Model

3 2𝑥 − 1 − 2(2𝑥 − 3)

3 2𝑎 + 3 + 2(𝑎 − 2)

Expand the brackets in these expressions.

Then simplify by collecting like terms:

© Copyright text
Quick Check

A

B

D

C

5𝑡 − 20

−5𝑡 − 20

−5𝑡 + 20

15𝑡

Expand:

−5 𝑡 − 4

© Copyright text
Quick Check

A

B

D

C

5𝑡 − 20 + 6𝑡 + 1

5𝑡 + 1 − 3𝑡 + 2

5𝑡 + 20 + 6𝑡 + 1 5𝑡 + 20 + 6𝑡 + 1

Expand:

5 𝑡 − 4 + 3 2𝑡 + 1

© Copyright text
Quick Check

A

B

D

C

𝑡 + 21

11𝑡 − 19

11𝑡 − 21

11𝑡 + 21

Simplify:

5𝑡 − 20 + 6𝑡 + 1

© Copyright text
Quick Check

A

B

D

C

14𝑥 + 4

14𝑥 + 3

12𝑥 − 6 + 2𝑥 + 10

14𝑥 − 14

Expand and simplify:

3 4𝑥 − 2 + 2(𝑥 + 5)

Talk Task

I think you should do:

𝑥 − 2 + 𝑥 − 2 + 𝑥 − 2 + 2𝑥 + 1 + 2𝑥 + 1

I think you should do:

3 𝑥 − 2 + 2 2𝑥 + 1

𝑥 − 2

𝑥 − 2

𝑥 − 2

2𝑥 + 1

2𝑥 + 1

Phil and Cala are trying to write an expression for the

total distance around this pentagon.

Who do you agree with?

Write the distance around the pentagon in a different way.

Talk Task - ANSWERS

I think you should do

𝑥 − 2 + 𝑥 − 2 + 𝑥 − 2 + 2𝑥 + 1 + 2𝑥 + 1

I think you should do
3 𝑥 − 2 + 2 2𝑥 + 1

Both students are correct.
3 𝑥 − 2 means three lots of (𝑥 − 2) which is the same as 𝑥 − 2 + 𝑥 − 2 + 𝑥 − 2
2(2𝑥 + 1) means two lots of (2𝑥 + 1) which is the same as 2𝑥 + 1 + 2𝑥 + 1

Some other ways of writing the expression for the distance around the
perimeter are:

3𝑥 − 6 + 4𝑥 + 2
7𝑥 − 4

Purposeful Practice

Use the cards to fill the gaps.

Simplify the expression.

(6 + 2𝑎)

Can you arrange the cards so that your answer is a constant?

Can you arrange the cards so you have no constant in your answer?

3𝑎 − 4 =

Purposeful Practice - ANSWERS

+ 2 (6 + 2𝑎)

−3 (3𝑎 − 4)

=

+12 + 4𝑎

−9𝑎 + 12

= −5𝑎 + 24

+ 2 (6 + 2𝑎)

−𝑎 (3𝑎 − 4)

=

+12 + 4𝑎

−3𝑎2+ 4𝑎

=12 + 8𝑎 − 3𝑎2

− 2 (6 + 2𝑎)

+3 (3𝑎 − 4)

=

−12 − 4𝑎

+9𝑎 − 12

=

5𝑎 − 24

− 2 (6 + 2𝑎)

+𝑎 (3𝑎 − 4)

=

−12 − 4𝑎

+3𝑎2− 4𝑎

=3𝑎2− 8𝑎 − 12

+ 3 (6 + 2𝑎)

−2 (3𝑎 − 4)

=

+18 + 6𝑎

−6𝑎 + 8

=

24

+ 3 (6 + 2𝑎)

−𝑎 (3𝑎 − 4)

=

+18 + 6𝑎

−3𝑎2+ 4𝑎

= 10𝑎 + 18 − 3𝑎2

− 3 (6 + 2𝑎)

+2 (3𝑎 − 4)

=

−18 − 6𝑎

+6𝑎 − 8

=

−26

− 3 (6 + 2𝑎)

+𝑎 (3𝑎 − 4)

=

−18 − 6𝑎

+3𝑎2− 4𝑎

= 3𝑎2− 10𝑎 − 18

+ 𝑎 (6 + 2𝑎)

−2 (3𝑎 − 4)

=

+6𝑎 + 2𝑎2

−6𝑎 + 8

=

8 + 2𝑎2

+ 𝑎 (6 + 2𝑎)

−3 (3𝑎 − 4)

=

+6𝑎 + 2𝑎2

−9𝑎2+ 12

=6𝑎 − 7𝑎2+ 12

− 𝑎 (6 + 2𝑎)

+2 (3𝑎 − 4)

=

−6𝑎 − 2𝑎2

+6𝑎 − 8

=

−8 − 2𝑎2

− 𝑎 (6 + 2𝑎)

+3 (3𝑎 − 4)

=

−6𝑎 − 2𝑎2

+9𝑎2− 12

=

9𝑎2− 12

© Copyright text
Ready to go?

Expand and simplify:

2 𝑥 + 3 + 3(2𝑥 − 1)

Exit Ticket

1) Use the grids to help you expand the brackets in each expression.
Then simplify by collecting like terms.

a) 4 𝑥 + 3 + 2(4 − 5𝑥)

b) 5(8𝑝 + 6) − 3(2𝑝 − 4)

×

𝑥

+3

4

×

4

−5𝑥

+2

×

8𝑝

+6

5

×

2𝑝

−4

−3

Exit Ticket ANSWERS

1) Use the grids to help you expand the brackets in each expression.
Then simplify by collecting like terms.

a) 4 𝑥 + 3 + 2(4 − 5𝑥)

b) 5(8𝑝 + 6) − 3(2𝑝 − 4)

×

𝑥

+3

4

4𝑥

+12

×

4

−5𝑥

+2

+8

−10𝑥

×

8𝑝

+6

5

40𝑝

+30

×

2𝑝

−4

−3

−6𝑝

+12

= 4𝑥 + 12 + 8 − 10𝑥
= −6𝑥 + 20

= 40𝑝 + 30 − 6𝑝 + 12
= 34𝑝 + 42

Students will:

Example

understand that a factor is a quantity that can be divided into a
term exactly
3 is a factor of 6

understand that factors can be numerical as well as algebraic
𝑥 is a factor of 6𝑥 (6𝑥 ÷ 𝑥 = 6)

3𝑥 is a factor of 6𝑥 (6𝑥 ÷ 3𝑥 = 2)

understand that the highestcommonfactor can be a combination
of letters and numbers

The HCF of 4𝑎 and 12 is 4

The HCF of 4𝑎2 and 12𝑎 is 4𝑎

be able to identify factors of a number, including algebraic factors
List all the factors of 5𝑎𝑏
List all the factors of 21𝑝𝑞

be able to identify the highest common factor of algebraic terms
What are the common factors of 3𝑎𝑏 and 9𝑏𝑐?

What is the HCF of 8𝑥2 and 6𝑥?

Lesson 9: Algebraic Factors

Unit

Example

Y7U2: Properties of
arithmetic

Distributivity – expanding brackets with
numbers

Y7U3: Factors and
multiples
Factors and factor pairs

Y7U4: Prime Factor
Decomposition
Highest common factors

Prior learning:

Learning objectives:

Retrieve

One of the factors of this

number is 58

This number has exactly

5factors

This is the smallest

number with 4 and 6 as

factors.

What numbers could the cards be describing?

Could the cards be describing more than one number?

These cards describe different numbers less than100.

Retrieve - ANSWERS

58

(1 × 58)

16

(1, 2, 4, 8, 16)

81

(1, 3,9, 27,81)
12

One of the factors of this

number is 58

This number has exactly

5 factors

This is the smallest

number with 4 and 6 as

factors.

Model

15

A factor divides exactly into a term.

Factors can be algebraic (letters) as well as numerical (numbers).

𝑎𝑏𝑐

2𝑥

List all the factors of:

A

© Copyright text
Quick Check

A

B

D

C

3

4

5

6

Which of these is a factor of 6?

© Copyright text
Quick Check

A

B

D

C

2

𝑦

1

4𝑦

Which of these is not a factor of 2𝑦?

© Copyright text
Ready to go?

List all the factors of:

10𝑎

A

Explain

5 divides exactly
into 15 and 5𝑏.

The factors of 15 are:

1

3

5

15

The factors of 5𝑏 are:

1

5

𝑏

5𝑏

When terms in an expression have the same factor,

this is called a common factor.

The common factors of 15 and 5𝑏 are 1 and 5.

common

factor

Model

Common factors can be algebraic.

Find the common factors of:

The common factors of

4𝑥 and 𝑥3 are …….

4𝑥 and 𝑥3

10𝑎 and 2𝑎𝑏

… divides exactly
into 10𝑎 and 2𝑎𝑏.

B

© Copyright text
Quick Check

A

B

D

C

1 and 3

1, 3, and 𝑦

9𝑦

1, 3, 𝑦 and 3𝑦

What are the common factors of 15 and 5𝑦?

The factors of 9𝑦 are:

1,

3,

9,

𝑦, 3𝑦,

9𝑦

The factors of 3𝑦 are:

1,

3, 𝑦,

3𝑦

© Copyright text
Ready to go?

B

What are the common factors of 8𝑎𝑏 and 12𝑎?

Model

The highest common factor in algebra is

the greatest number you divide by and any common variables.

Find the highest common factor of:

8𝑥 and 4𝑥3

10𝑎𝑏 and 4𝑎

The highest common

factor of 10𝑎𝑏 and 4𝑎 is …

C

© Copyright text
Ready to go?

What is the highest common factor of 15𝑎𝑏 and 20𝑏?

C

Purposeful Practice

3𝑎 has exactly four factors:

Find some other expression with exactly four factors.

What do you notice?

Can you find an algebraic expression with exactly three factors?

1

3

𝑎

3𝑎

What about if you

use negatives?

Extension prompt

2

What if you use
more than one

letter?

Extension prompt

1

Purposeful Practice - ANSWERS

Any prime number multiplied by a variable will have exactly four factors:

e.g. 2𝑟: 1, 2, 𝑟, 2𝑟

Any two variables multiplied together will have exactly four factors:

e.g. 𝑎𝑏: 1, 𝑎, 𝑏, 𝑎𝑏

It is impossible to find an algebraic expression with exactly three factors.
However, you can find a numerical expression with exactly three factors.
These are the square numbers:

e.g. 4: 1, 2, 4

Exit Ticket

1. List all the common factors of:

a) 12 and 15

b) 6𝑥 and 9𝑥

2. Write down the highest common factor of:

a) 3𝑥 and 9

b) 8𝑎 and 12𝑏

c) 𝑦2 and 𝑦

d) 10𝑝𝑞and15𝑝𝑞𝑟

e) 4𝑐𝑑 and 2𝑐2

Exit Ticket ANSWERS

1. List all the common factors of:

a) 12 and 15

b) 6𝑥 and 9𝑥

2. Write down the highest common factor of:

a) 3𝑥 and 9

b) 8𝑎 and 12𝑏

c) 𝑦2 and 𝑦

d) 10𝑝𝑞and15𝑝𝑞𝑟

e) 4𝑐𝑑 and 2𝑐2

a) 1 and 13

b) 1, 3, 𝑥

a) 3

b) 4

c) 𝑦

d) 5𝑝𝑞

e) 2𝑐

Students will:

Example

understand that terms in expressions can have common factors

3𝑥 + 6 can be written as 3 × 𝑥 + 3 × 2

understand that factorising means writing an expression as a
multiple of a pair of brackets (the opposite of expanding brackets).
Factorising 3𝑥 + 6 means writing it as 3(𝑥 + 2)

understand that to fully factorise, the multiple of the brackets is the
highestcommonfactor of the terms in the expression
12𝑏 + 18 = 6(2𝑏 + 3) (rather than 2 6𝑏 + 9 )

be able to factorise an expression with a common numerical factor

Factorise 4𝑎 − 20

be able to factorise an expression with a common algebraic factor

Factorise 5𝑎 − 15𝑎2

Lesson 10: Factorising expressions into a single pair of brackets

Unit

Example

Y7U2: Properties of
arithmetic

Distributivity – expanding brackets with
numbers

Y7U3: Factors and
multiples
Factors and factor pairs

Y7U4: Prime Factor
Decomposition
Highest common factors

Prior learning:

Learning objectives:

Retrieve

What’s the same about these expressions?

What’s different?

1(12𝑛 − 24)

2(6𝑛 − 12)

3(4𝑛 − 8)

4(3𝑛 − 6)

Write another expression that could be part of this group.

Retrieve

1(12𝑛 − 24)

2(6𝑛 − 12)

3(4𝑛 − 8)

4(3𝑛 − 6)

The same:

They all have a number multiplied by
a pair of brackets. The number is
called the coefficient of the brackets.

When the brackets are expanded
they all equal 12𝑛 − 24

Different:

The coefficient of the brackets is
different in each expression.

The terms inside the brackets are
different.

Other expressions that expand to 𝟏𝟐𝐧 − 𝟐𝟒:

e.g. 6(2𝑛 − 4)12(𝑛 − 2)24(

1
2𝑛 − 1)0.5(24𝑛 − 48)−2(12 − 6𝑛)

Explain

Factorising means writing an expression as

a multiple of a pair of brackets.

It is the opposite of expanding brackets.

2 3𝑥 − 4 = ………………
… ( ………… )= 10𝑥 + 4

𝑥
−1
𝑥𝑥
−1

−1 −1

𝑥
−1
𝑥𝑥
−1

−1 −1

𝑥

1

𝑥𝑥

1

11

𝑥𝑥𝑥

𝑥𝑥

𝑥𝑥

When we expand brackets we

multiply everything inside the brackets

by the coefficient of the brackets.

When we factorise we need to

divide every term in the expression
by the coefficient of the brackets.

Explain

10𝑥 + 15 = … ( ……………)

To factorise an expression, you need to find the
common factors of the terms in the expression.

The highest

commonfactor of

10𝑥 and 15 is …

×

10𝑥

+15

𝑥

𝑥

1

1

1

𝑥

𝑥

1

1

1

𝑥

𝑥

1

1

1

𝑥

𝑥

1

1

1

𝑥

𝑥

1

1

1

Model

Factorise 14 − 21𝑐
Factorise 6𝑎 − 9𝑏

A

𝑎

𝑎

−𝑏

−𝑏

−𝑏

𝑎

𝑎

−𝑏

−𝑏

−𝑏

𝑎

𝑎

−𝑏

−𝑏

−𝑏

© Copyright text
Quick Check

A

B

D

C

1

11

2
There are no

common factors

What are the common factors of:

22𝑦 and 33

?

© Copyright text
Ready to go?

Factorise :

4𝑑 + 6𝑠

Model

To fully factorise means to factorise with the highest common factor.

The highest common factor can be algebraic.

Fully factorise 15𝑥 − 20𝑥2

Fully factorise 12𝑎𝑏 + 8𝑏

We divide each term by the

highestcommonfactor to work
out what goes inside the bracket

What is the highest
common factor of

15𝑥 and 20𝑥2?

B

© Copyright text
Quick Check

A

B

D

C

2

4

1

84

What is the highest common factor of:

28𝑎 + 12

© Copyright text
Quick Check

A

B

D

C

8

2

5

4

What is the missing number?

×

2𝑚

−5

?
8𝑚

−20

© Copyright text
Quick Check

A

B

D

C

4.5

−3

3

−4.5

What is the missing number?

×

2𝑚
?

3

6𝑚

−9

© Copyright text
Quick Check

A

B

D

C

8(2𝑏 − 0)

8𝑏

8(2𝑏 − 1)

4(4𝑏 − 2)

Factorise fully:

16𝑏 − 8

Talk Task

The Maths Mastery students are trying to fully factorise this expression:

24𝑎𝑏 − 12𝑎

I think 24𝑎𝑏 − 12𝑎 fully

factorised is:

6𝑎(4𝑏 − 2)

I think 24𝑎𝑏 − 12𝑎 fully

factorised is:

12𝑎(2𝑏)

Where have the students gone wrong?

Talk Task - ANSWERS

I think 24𝑎𝑏 − 12𝑎 fully

factorised is:

6𝑎(4𝑏 − 2)

I think 24𝑎𝑏 − 12𝑎 fully

factorised is:

12𝑎(2𝑏)

The expression is only partially
factorised because 12 is the
highest common factor, not 6

−12𝑎 ÷ 12𝑎 = −1.

Declan has missed the −1 out
of his brackets, thinking that

− 12𝑎 ÷ 12𝑎 = 0

© Copyright text
Ready to go?

Fully factorise the expression:

16𝑚 + 12

Purposeful Practice

In how many different ways can you factorise:

24𝑎𝑏 − 12𝑎

In how many different ways can you fully factorise it?

Could you use

decimals?

Extension prompt

3

What about

fractions?

Extension prompt

2

What if you use

negative
numbers?

Extension prompt

1

Purposeful Practice

24𝑎𝑏 − 12𝑎 can be factorised as:

𝑎(12𝑎 − 6)

2𝑎(12𝑏 − 6𝑎)

3𝑎(8𝑏 − 4𝑎)

4𝑎(6𝑏 − 3𝑎)

6𝑎 4𝑏 − 2𝑎

12𝑎(2𝑏 − 𝑎)

1 24𝑎𝑏 − 12𝑎

2(12𝑎𝑏 − 6𝑎)

3(8𝑎𝑏 − 4𝑎)

4(6𝑎𝑏 − 3𝑎)

6 4𝑎𝑏 − 2𝑎

12(2𝑎𝑏 − 𝑎)

Negatives and fractions can also be used to find expressions that expand to

24𝑎𝑏 − 12𝑎:

e.g. −6(−4𝑎𝑏 + 2𝑎)0.5(48𝑎𝑏 − 34𝑎)−3𝑎(4𝑎 − 8𝑏)

Exit Ticket

1. Find the highest common factor of:

a) 3𝑎 + 12

b) 6 − 18𝑏

c) 𝑦2+ 2𝑦

2. Use your answers from question 1 to help you factorise each expression.

×

3𝑎

12

×

6

−18𝑏

×

𝑦2

+2𝑦

3𝑎 + 12 = ____________

6 − 18𝑏 = ____________

𝑦2+ 2𝑦 = ____________

Exit Ticket ANSWERS

1. Find the highest common factor of:

a) 3𝑎 + 12

b) 6 − 18𝑏

c) 𝑦2+ 2𝑦

2. Use your answers from question 1 to help you factorise each expression.

×

3𝑎

12

×

6

−18𝑏

×

𝑦2

+2𝑦

3𝑎 + 12 = ____________

6 − 18𝑏 = ____________

𝑦2+ 2𝑦 = ____________

a) 3

b) 6

c) 𝑦

3

𝑎

+4

6

1

−3𝑏

𝑦

𝑦

+2

3(𝑎 + 4)

6(1 − 3𝑏)

𝑦(𝑦 + 2)